%Macros
\newtheorem{lemma}{Lemma}
\newtheorem{theorem}[lemma]{Theorem}
\newtheorem{informaltheorem}[lemma]{Informal Theorem}
\newtheorem{informallemma}[lemma]{Informal Lemma}
\newtheorem{corollary}[lemma]{Corollary}
\newtheorem{definition}[lemma]{Definition}
\newtheorem{proposition}[lemma]{Proposition}
\newtheorem{question}{Question}
\newtheorem{problem}{Problem}
\newtheorem{remark}[lemma]{Remark}
\newtheorem{claim}{Claim}
\newtheorem{fact}{Fact}
\newtheorem{challenge}{Challenge}
\newtheorem{observation}{Observation}
\newtheorem{openproblem}{Open Problem}
\newtheorem{openquestion}{Open question}
\newtheorem{conjecture}{Conjecture}
\newtheorem{todo}{ToDo}

\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\beas}{\begin{eqnarray*}}
\newcommand{\eeas}{\end{eqnarray*}}

\newcommand{\poly}{\mathrm{poly}}
\newcommand{\eps}{\epsilon}
\newcommand{\e}{\epsilon}
\newcommand{\polylog}{\mathrm{polylog}}
\newcommand{\rob}[1]{\left( #1 \right)} %Round Brackets
\newcommand{\sqb}[1]{\left[ #1 \right]} %square Brackets
\newcommand{\cub}[1]{\left\{ #1 \right\} } %curly brackets
\newcommand{\rb}[1]{\left( #1 \right)} %Round
\newcommand{\abs}[1]{\left| #1 \right|} %| |
\newcommand{\zo}{\{0, 1\}}
\newcommand{\zonzo}{\zo^n \to \zo}
\newcommand{\zokzo}{\zo^k \to \zo}
\newcommand{\zot}{\{0,1,2\}}

\newcommand{\en}[1]{\marginpar{\textbf{#1}}}
\newcommand{\efn}[1]{\footnote{\textbf{#1}}}

\newcommand{\junk}[1]{}

\newcommand{\dsumprob}[2]{P(#1,#2)}
\newcommand{\prob}[1]{\Pr \left[ #1 \right]}
\newcommand{\dprob}[2]{\Pr \left[ #1 \mbox{;} #2 \right]}
\newcommand{\expect}[1]{\mathbb{E}\left[ #1 \right]}
\newcommand{\mindegree}{\delta}

\newcommand{\md}[1]{\delta_{#1}}                            % min degree
\newcommand{\gf}[1]{G_{#1}}                                     % graph
\newcommand{\nei}[3]{N^{#2}_{#1}\rb{#3}}                % neighbor
\newcommand{\dg}[2]{d_{#1}\rb{#2}}                          % degree
\newcommand{\dgi}[3]{d_{#1}\rb{#2,#3}}                   % induced degree
\newcommand{\prb}[2]{p_{#1,#2}}

\newcommand{\cost}{\mbox{cost}}
\newcommand{\GNS}[1]{\mbox{{\sc GNS}}(#1)}
\newcommand{\BfPara}[1]{\noindent {\bf #1}.}

\newcommand{\cc}{c}
\newcommand{\cp}{c'}
 
\newcommand{\pt}{p}
\newcommand{\pf}{p_f}
\newcommand{\ps}{p_s}
\newcommand{\pv}{p_v}
\newcommand{\pb}{p_m} 
\newcommand{\vs}{\mbox{{\sc VS}}}
\newcommand{\vf}{\mbox{{\sc VF}}}
\newcommand{\uv}{\mbox{{\sc UV}}}

%End macros
